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Solve-The-Equation-7x-1-9x-1-21x-1-63x-1-160-189-

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Question Number 210858 by hardmath last updated on 20/Aug/24
Solve The Equation: (7x+1)(9x+1)(21x+1)(63x+1)= ((160)/(189))
$$\mathrm{Solve}\:\mathrm{The}\:\mathrm{Equation}: \\ $$$$\left(\mathrm{7x}+\mathrm{1}\right)\left(\mathrm{9x}+\mathrm{1}\right)\left(\mathrm{21x}+\mathrm{1}\right)\left(\mathrm{63x}+\mathrm{1}\right)=\:\frac{\mathrm{160}}{\mathrm{189}} \\ $$
Answered by mm1342 last updated on 20/Aug/24
(63x+9)(63x+7)(63x+3)(63x+1)=160 63x+5=u ⇒(u+4)(u+2)(u−2)(u−4)=160 (u^2 −16)(u^2 −4)=160 u^4 −20u^2 −96=0 u^2 =10±14⇒(63x+5)^2 =10±14 63x+5=±2(√6) or 63x+5=±2i ⇒x=((−5±2(√6))/(63)) or x=((−5±2i)/(63))
$$\left(\mathrm{63}{x}+\mathrm{9}\right)\left(\mathrm{63}{x}+\mathrm{7}\right)\left(\mathrm{63}{x}+\mathrm{3}\right)\left(\mathrm{63}{x}+\mathrm{1}\right)=\mathrm{160} \\ $$$$\mathrm{63}{x}+\mathrm{5}={u} \\ $$$$\Rightarrow\left({u}+\mathrm{4}\right)\left({u}+\mathrm{2}\right)\left({u}−\mathrm{2}\right)\left({u}−\mathrm{4}\right)=\mathrm{160} \\ $$$$\left({u}^{\mathrm{2}} −\mathrm{16}\right)\left({u}^{\mathrm{2}} −\mathrm{4}\right)=\mathrm{160} \\ $$$${u}^{\mathrm{4}} −\mathrm{20}{u}^{\mathrm{2}} −\mathrm{96}=\mathrm{0} \\ $$$${u}^{\mathrm{2}} =\mathrm{10}\pm\mathrm{14}\Rightarrow\left(\mathrm{63}{x}+\mathrm{5}\right)^{\mathrm{2}} =\mathrm{10}\pm\mathrm{14} \\ $$$$\mathrm{63}{x}+\mathrm{5}=\pm\mathrm{2}\sqrt{\mathrm{6}}\:\:\:{or}\:\:\:\mathrm{63}{x}+\mathrm{5}=\pm\mathrm{2}{i} \\ $$$$\Rightarrow{x}=\frac{−\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}}{\mathrm{63}}\:\:\:{or}\:\:\:{x}=\frac{−\mathrm{5}\pm\mathrm{2}{i}}{\mathrm{63}} \\ $$$$ \\ $$
Commented by hardmath last updated on 20/Aug/24
thank you dear professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$
Answered by Rasheed.Sindhi last updated on 20/Aug/24
7(x+(1/7)).9(x+(1/9)).21(x+(1/(21))).63(x+(1/(63)))=((160)/(189)) 63^2 .21{(x+(1/7))(x+(1/(63)))}{(x+(1/9))(x+(1/(21)))}=((160)/(189)) 63^2 .21(x^2 +((10)/(63))x+(1/(441)))(x^2 +((10)/(63))x+(1/(189)))=((160)/(189)) x^2 +((10)/(63))x=y 63^2 .21(y+(1/(441)))(y+(1/(189)))=((160)/(189)) 63^2 .21(y^2 +((10)/(1323))y+(1/(83349)))=((160)/(189)) 83349y^2 +630y+1=((160)/(189)) 15752961y^2 +119070y+29=0 y=((−119070±(√(119070^2 −4(15752961)(29))))/(2(15752961))) y=((−119070±111132)/(31505922))=−(1/(3969)),−((29)/(3969)) x^2 +((10)/(63))x=−(1/(3969)),−((29)/(3969)) { ((3969x^2 +630x+1=0)),((3969x^2 +630x+29=0)) :} { ((x=((−5±2(√6))/(63)) )),((x=((−5±2i)/(63)))) :}
$$\mathrm{7}\left({x}+\frac{\mathrm{1}}{\mathrm{7}}\right).\mathrm{9}\left({x}+\frac{\mathrm{1}}{\mathrm{9}}\right).\mathrm{21}\left({x}+\frac{\mathrm{1}}{\mathrm{21}}\right).\mathrm{63}\left({x}+\frac{\mathrm{1}}{\mathrm{63}}\right)=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$$\mathrm{63}^{\mathrm{2}} .\mathrm{21}\left\{\left({x}+\frac{\mathrm{1}}{\mathrm{7}}\right)\left({x}+\frac{\mathrm{1}}{\mathrm{63}}\right)\right\}\left\{\left({x}+\frac{\mathrm{1}}{\mathrm{9}}\right)\left({x}+\frac{\mathrm{1}}{\mathrm{21}}\right)\right\}=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$$\mathrm{63}^{\mathrm{2}} .\mathrm{21}\left({x}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{63}}{x}+\frac{\mathrm{1}}{\mathrm{441}}\right)\left({x}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{63}}{x}+\frac{\mathrm{1}}{\mathrm{189}}\right)=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$${x}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{63}}{x}={y} \\ $$$$\mathrm{63}^{\mathrm{2}} .\mathrm{21}\left({y}+\frac{\mathrm{1}}{\mathrm{441}}\right)\left({y}+\frac{\mathrm{1}}{\mathrm{189}}\right)=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$$\mathrm{63}^{\mathrm{2}} .\mathrm{21}\left({y}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{1323}}{y}+\frac{\mathrm{1}}{\mathrm{83349}}\right)=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$$\mathrm{83349}{y}^{\mathrm{2}} +\mathrm{630}{y}+\mathrm{1}=\frac{\mathrm{160}}{\mathrm{189}} \\ $$$$\mathrm{15752961}{y}^{\mathrm{2}} +\mathrm{119070}{y}+\mathrm{29}=\mathrm{0} \\ $$$${y}=\frac{−\mathrm{119070}\pm\sqrt{\mathrm{119070}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{15752961}\right)\left(\mathrm{29}\right)}}{\mathrm{2}\left(\mathrm{15752961}\right)} \\ $$$${y}=\frac{−\mathrm{119070}\pm\mathrm{111132}}{\mathrm{31505922}}=−\frac{\mathrm{1}}{\mathrm{3969}},−\frac{\mathrm{29}}{\mathrm{3969}} \\ $$$${x}^{\mathrm{2}} +\frac{\mathrm{10}}{\mathrm{63}}{x}=−\frac{\mathrm{1}}{\mathrm{3969}},−\frac{\mathrm{29}}{\mathrm{3969}} \\ $$$$\begin{cases}{\mathrm{3969}{x}^{\mathrm{2}} +\mathrm{630}{x}+\mathrm{1}=\mathrm{0}}\\{\mathrm{3969}{x}^{\mathrm{2}} +\mathrm{630}{x}+\mathrm{29}=\mathrm{0}}\end{cases}\: \\ $$$$\begin{cases}{{x}=\frac{−\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}}{\mathrm{63}}\:}\\{{x}=\frac{−\mathrm{5}\pm\mathrm{2}\boldsymbol{{i}}}{\mathrm{63}}}\end{cases} \\ $$
Commented by hardmath last updated on 20/Aug/24
thank you dear professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$

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